Curves: how to make circular shapes within curves
You are missing that parametric curves have nothing to do with spheres.


Screenshoted frmo the preview of An accurate NURBS curve interpolation algorithm with short spline interpolation capacity

After you are satisfied with your curve you can utilize the To Sphere command. However, since it operate on points only, you have to do it after adding more points.


Duplicate your curve.
Instead of subdiving, remove the bevel, so the curve is only a curve. Then press AltC > Convert to mesh to convert it to a mesh. The points on the curve are now sampled. Convert is back to a curve with AltC > Convert to curve. Re-add your bevel settings.
If the center of your curve is at the world origin, move the 3D cursor to the origin and set it as the pivot point NumpadPeriod.
Select all. A
Execute the to sphere command. &#9096; Ctrl&#8679; Shift&#9095; AltS